We calculated the statistical uncertainty in \(X_{\rm LAH}\) . Statistical uncertainties for fractions (i.e., Bernoulli trials) are given by a binomial proportion confidence interval (BPCI). Following our previous paper on LAE fractions [1]}, we derived the statistical uncertainties in \(X_{\rm LAH}\)  using astropy.stats.binom_conf_interval, with the Wilson score interval as an approximation formula [2]}. Our \(1\sigma \)  uncertainties correspond to 68.3% confidence intervals. For samples including RID=6693 or 54891, which has a low ZCONF=0, we adopted fractions including them in the \(\mbox{Ly$\alpha $}\)  fractions but took conservative uncertainties consisting of the minimum and maximum values of 68.3% confidence intervals of fractions with and without RID=6693 or 54891.
